Adaptive Backstepping Sliding Mode Fault Tolerant Control for Satellite Attitude under Actuator Faults

Modern spacecraft missions place stringent performance requirements on the reliability and stability of spacecraft attitude control system. However, during long-term in-orbit mission, actuators may suffer faults. In this paper, we proposed an adaptive fault tolerant control law for attitude control under actuator failures and external disturbances. Firstly, the attitude dynamic model under actuator faults and attitude kinematics are described. Then, in order to solve the attitude control problem under actuator faults and external disturbances, an adaptive backstepping sliding mode fault tolerant control scheme is designed. According to Lyapunov theory, the closed-loop system is proved to be globally asymptotically stable. Finally, the numerical simulation results demonstrate that the proposed adaptive controller can achieve anticipative attitude control performance, fault information estimation and well fault-tolerant capability.


Introduction
The growing development of different space exploration missions has motivated significant demands for the reliable and autonomous control of spacecraft. Attitude control system (ACS), as one of spacecraft's subsystems, plays a valuable role in guaranteeing the success of space missions. During the past decades, far-ranging efforts have been devoted to this issue, and a large amount of modern control techniques are documented in the literature [1][2][3][4][5]. However, most of these existing methods focus on spacecraft with healthy actuators. During the long-term in-orbit mission, the possibility of malfunction in actuator increases due to spacecraft's challenging operating conditions. Once the anomaly do occur, there would be huge economy loss. Hence, how to guarantee the reliability and safety of the controlled plants has become a serious problem to be solved.
Generally speaking, the existing fault tolerant control (FTC) methods can be divided into two parts: passive FTC and active FTC. The passive FTC [6,7] is based on the robust technique and implemented as a fixed controller. However, the passive FTC only tolerates a limited number of faults, and it achieves robustness at the cost of decreased nominal performance. In contrast, active FTC solves the fault control problem by adjusting and modifying the controller based on real-time fault detection, identification and controller reconfiguration so that the stability of system and well control performance can be maintained. Taking advantage of greater performance and ability to handle a wider range of failure, the active FTC has motivated more significant research during the past years, such as multiple model [8], feedback linearization [9], sliding mode control [10], adaptive control [11,12], control allocation [13,14], eigenstructure assignment [15], model predictive control [16,17] [10,11] can only deal with partial failure of actuator. Therefore, the fault tolerance of these control methods is very limited. On the other hand, spacecraft in-orbit will inevitably be affected by external disturbances such as gravity gradient moment, aerodynamic moment, etc. This problem will limit the practical application of the existing attitude fault tolerant control methods. External disturbance and actuator fault are two key problems which affect the performance of fault tolerant controller. For this reason, how to design attitude fault tolerant controller to realize reliable and stable control under the condition of multi-type failure of actuator and external disturbance has become an urgent problem to be solved.
In this paper, we concentrate on the design of an active fault tolerant control algorithm for spacecraft attitude under multiple faults and external disturbances. Based on the backstepping control algorithm, the sliding mode was introduced because of its strong robustness. The system change caused by fault can be regarded as an uncertainty, and this uncertainty can be estimated by adaptive algorithm and used for controller design. Simulation results illustrate the well performance of the adaptive backstepping sliding mode fault tolerant controller.
The rest of this paper is organized as follows. Section 2 is devoted to describe the attitude kinematics and attitude dynamics with unknown actuator faults. In Section 3, an adaptive fault tolerant controller is proposed and closed-loop system stability analysis is derived. The effectiveness and superiority of the control scheme are verified by numerical simulation in Section 4. Finally, the conclusion of this paper is shown in Section 5.

Attitude dynamics
The satellite attitude dynamics is ̇ where denotes the inertia matrix of the satellite; is the satellite attitude velocity; is the control torque; is the external disturbance torque.
For a healthy-actuator system, the actual output can accurately respond to the command input, . However, actuator faults are usually inevitable. These faults include: 1) Partial loss of energy: actuator can only respond partially to the expected control command; 2) Continuous float: actuator cannot output the expected actuating power accurately, and there exists a small bias; 3) Complete loss of energy: actuator completely fails to work; 4) Locking: actuator has a fixed and uncontrolled output. In this study, loss of effectiveness fault and bias fault are considered. where is quaternion product. The error angular velocity is written as where is the initial angular velocity; is the desired angular velocity. Substituting Eq.6 and Eq.7 into Eq.4 and Eq.5, and the error quaternion kinematics can be derived as ̇

Fault tolerant controller design
In this section, the adaptive backstepping sliding mode fault tolerant controller is proposed to deal with the attitude control problem under the actuator faults and external disturbances.
The first-order backstepping variable is The second-order virtual backstepping variable is There exists shaking in the sliding mode control, and it can be weakened by introducing the approach law. In order to guarantee system robustness and reduce the shaking caused by external disturbances, the exponential approach law is adopted in this paper. In order to make ̇ , the controller can be designed as follows where ̂ and ̂ are the estimation of and , respectively. We define the estimation error as and , and thus we have ̂ ̂ Substituting the proposed controller Eq.22 into Eq.21, and combining Eq.23 and Eq.24, after some simplification, we get  ; ; ; the actuators partial loss of effectiveness matrix is chosen as ; the bias fault matrix is . . Control torque (command and actual) Applying the controller Eq.22 and adaptive update laws Eq.27 and Eq.28, the simulation results are illustrated as Fig.1-5. The time responses of the attitude and attitude velocity are presented in Fig.1-2. The satellite attitude and attitude velocity converge to zero asymptotically under the adaptive fault tolerant controller. Evolution of the estimations of actuator faults are displayed by Fig.3-4. It can be seen that the estimated loss of effectiveness and bias fault are both bounded and converge to the true value in finite time. Fig.5 displays the comparison between the command control torque and actual control torque. The adaptive controller can adjust the actual control output according to the estimated fault to achieve better control performance.

Conclusion
In this paper, an adaptive fault tolerant controller for satellite attitude under actuator faults and external disturbances is presented. The loss of effectiveness and bias fault are considered in the actuators. The adaptive controller is designed by employing backstepping and sliding mode technique. According to the Lyapunov theory, the proposed controller ensures that the closed-loop system is globally ultimately uniformly bounded. Numerical simulation results demonstrate that the proposed